computational design scheme for wind turbine drive-train
TRANSCRIPT
Research ArticleComputational Design Scheme for Wind Turbine Drive-TrainBased on Lagrange Multipliers
Mohammed Saleh, Ayman Nada, Ahmed El-Betar, and Ahmed El-Assal
Benha Faculty of Engineering, Benha University, Benha 13512, Egypt
Correspondence should be addressed to Ayman Nada; [email protected]
Received 5 April 2017; Accepted 20 June 2017; Published 21 November 2017
Academic Editor: Thordur Runolfsson
Copyright ยฉ 2017 Mohammed Saleh et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
The design optimization of wind turbines and their subsystems will make them competitive as an ideal alternative for energy. Thispaper proposed a design procedure for one of these subsystems, which is the Wind Turbine Drive-Train (WTDT). The designof the WTDT is based on the load assumptions and considered as the most significant parameter for increasing the efficiency ofenergy generation. In industry, these loads are supplemented by expert assumptions and manipulated to design the transmissionelements. In contrary, in thiswork, themultibody systemapproach is used to estimate the static aswell as dynamic loads based on theLagrangemultipliers. Lagrangemultipliers are numerical parameters associated with the holonomic and nonholonomic constraintsassigned in the drive-train model. The proposed scheme includes computational manipulations of kinematic constraints, mappingthe generalized forces into Cartesian respective, and enactment of velocity-based constrains. Based on the dynamic model and theobtained forces, the design process of a planetary stage of WTDT is implemented with trade-off โs optimization in terms of gearingparameters. A wind turbine of 1.4 megawatts is introduced as an evaluation study of the proposed procedure, in which the mainadvantage is the systematic nature of designing complex systems in motion.
1. Introduction
Wind turbine is a mechanical system utilized to convert windto electrical energy through high-speed generator. The sizeof commercial wind turbines has increased dramatically inthe last 25 years from approximately rated power of 50 [KW]and a rotor diameter of 10 to 15 [m] up to 8 [MW] witha rotor diameter of more than 140 [m] [1]. The mechani-cal power generated via turbine blades is characterized byhigh torque and low rotational speed of the rotor, between 5and 22 [rpm]. Thus, the next step is necessarily the design ofpower conversion system or drive-train that produces stan-dard electricity from the shaft power developed by the rotor.Most of wind turbines in operation field have the traditionalDanish design; that is, three blades are rigidly fixed to therotor, which is indirectly coupled with an electric generatorthrough a gearbox. Figure 1 shows the nacelle arrangementof such turbines including a high-speed induction generator,typically operated around 1000 to 2000 [rpm], in a clearlymodular arrangement.
The design optimization of such a turbine and its sub-systems will make future products more attractive comparedwith fossil and nuclear power plants. Instead of using thetorsional models of wind turbine to simulate and design thevarious components of wind turbines, it is proposed to imple-ment the multibody system dynamics approach [2, 3], todevelop a consistent model to describe correctly the dynamicbehavior of wind turbine blades and of the drive-train aswell.
Under the umbrella of the multibody system dynam-ics, efficient procedure was developed for mapping airfoilsgeometry of wind turbine blades into flexible models usingabsolute nodal coordinate formulation (ANCF) [4, 5]. Thisprocedure is extended to involve the complete aerodynamicaland structural behavior of blades with the nonuniform andtwisted configuration [6]. The effectiveness of using ANCFin modeling large-size wind turbine blades is examinedthrough dynamic/static simulation procedure, and a methodfor designing such blades within DFD process is introducedsuccessfully for practical wind conditions [7].
HindawiJournal of EnergyVolume 2017, Article ID 4027834, 16 pageshttps://doi.org/10.1155/2017/4027834
2 Journal of Energy
(1)
(2) (3)
(4) (5) (6) (7) (8) (9) (10)(11)
(12)
(13)
(14) (15) (16)(17) (18) (19) (20)
(1) Spinner(2) Spinner bracket(3) Blade(4) Pitch bearing(5) Rotor hub(6) Main bearing(7) Main sha๏ฟฝ(8) Gearbox(9) Brake disc(10) Coupling
(11) Generator(12) Service crane(13) Meteorological sensors(14) Tower(15) Yaw ring(16) Yaw gear(17) Nacelle bedplate(18) Oil filter(19) Canopy(20) Generator fan
Figure 1: Nacelle arrangement (reference: Siemens website).
Although the cost of the rotor blades is approximately20% of the total cost of wind turbine, it has a working lifetimeof twenty years. In contrast, the gearboxes are noticed to failordinarily within operational time interval of five years, andneed substitution. This is a costly task, since the substitutionof a gearboxes accounts for about 10% of the constructionand installation cost of the wind turbine. That will negativelyaffect the estimated income from a wind turbine.
Particular design procedures with the auxiliary functionof computer simulation on wind turbine gearbox design havebeen discussed in literature [8]. Instead of the torsional-based models of the wind turbine drive-train, which werestudied in many literatures [9โ12], the multibody modelenables obtaining reaction forces as well as reactionmomentsbetween the meshing gears and between gears and ground-structure.
This paper proposes a design procedure for wind turbinedrive-train (WTDT) based on multibody system approach.This scheme can be integrated with the previously introducedprocedures for modeling and design of wind blades [7]. Thecurrent work, which considers only rigid bodies, presents acomputational scheme for designing the wind turbine drive-train based on gear tooth stresses. The flexibility effects areignored assuming that they have little effect in estimating themaximum forces proposed for design process. Static as wellas dynamic loads are calculated based on the Lagrange mul-tipliers, which are numerical parameters associated with theholonomic and nonholonomic constraints assigned for thedrive-trainmodel.Thenumericalmanipulations of kinematic
stabilization, mapping the generalized forces into Cartesianrespective, and implementation of velocity-based constrainsinto quadratic terms are carried out, and the correspondingsubroutines are constructed. Based on the dynamic modeland the obtained forces, the design process of a standardgearbox with (at least) one planetary stage is demonstratedwith trade-offs optimization in terms of gearing parameters.
This paper is organized as follows: Section 2 presents thegeneral form of equations of motion for multibody systemand describes the relativemotion between the interconnectedbodies, that is, whether holonomic or nonholonomic type ofconstraints. Section 3 illustrates the proposed procedure andintroduces the various design modules. Furthermore, twosubsections are included to present the method of calculatingthe generalized aerodynamic forces and also the necessaryequations for stress analysis. In Sections 4 and 5, a case studyof practical wind turbine is manipulated using the proposedprocedure. Section 6 summarizes the concluding remarksand conclusion.
2. Multibody Dynamics
In contrast to the blade model, which is definitely defined asflexible body, the gearbox model is constructed by consider-ing only rigid bodies for its components. It can be shown thatrigid body can be considered as a special case of the floatingframe of reference formulation [13, 14], which can be usedin estimating the stress along certain elements after carryingout the simulation. In this case, the system of interconnected
Journal of Energy 3
bodies, the overall equation of motion can be written in amatrix form as
[M C๐๐C๐ 0
][q๐] = [Q
Q๐] , (1)
where M is the mass matrix and q are the generalizedcoordinates and expressed as q = [R๐ ๐๐]๐, such thatR = [๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง]๐ are the Cartesian coordinates of the bodyframe origin and ๐ = [๐1 ๐2 ๐3 ๐4]๐ is the vector of Eulerparameters that represent the rotational coordinates [15]. ๐is the vector of Lagrange multipliers and Q is the vector ofgeneralized forces. The vector Q๐ absorbs all terms that arequadratic in the velocity; it can be expressed as [2]
Q๐ = C๐q. (2)
The matrix C๐ is the kinematic constraints Jacobian matrix,where the constraint equations C(q, q, ๐ก) can be presented as
C (q, q, ๐ก) = [ Ch (q, ๐ก)Cnh (q, q, ๐ก)] = 0 (3)
such that Ch and Cnh are the holonomic and nonholonomicconstraint equations, respectively. If the nonholonomic con-straints are linear in the generalized velocities, they could beexpressed as follows [16]:
Cnh (q, q, ๐ก) = C (q, ๐ก) q + ๏ฟฝc (q, ๐ก) = 0, (4)
where the matrix C and vector ๏ฟฝc depend on the systemcoordinates and may depend on time as well. These types ofconstraints restrict the generalized velocities and accordinglyrestrict the generalized acceleration but do not, however,apply any constraint on the generalized coordinates. Conse-quently, the number of independent coordinates is greaterthan the number of independent velocity or accelerationvalues. It should be mentioned here that the nonholonomicconstraint is best suited to describe the relative motionbetween meshing gears; see Figure 2.
The Jacobian of constrained system corresponding toboth holonomic and nonholonomic types of constraints canbe expressed as
C(q,q) = [[[[
๐Ch (q, ๐ก)๐q๐Cnh (q, q, ๐ก)๐q]]]]= [Ch
q
Cnhq] = [ Ch
q
C (q, ๐ก)] , (5)
whereChq is the Jacobian matrix of the holonomic constraints
andCnhq = C(q, ๐ก) is the Jacobianmatrix of the nonholonomic
constraints. The time derivative of the constraints functioncan be written as
C (q, q, t) = [ Ch๐ q + Ch
๐ก
C (q, ๐ก) q + ๏ฟฝc (q, ๐ก)] = 0. (6)
c
Z
Y
X
uc
Rc
z1 x1
y1
rc
Figure 2: Multibody model of epicyclic gear train, showing onlyone body frame and displacement field of point ๐; the holonomicconstraints include the rigid joint between the ring gear and groundand revolute joints between gears and ground. The nonholonomicconstraints include the zero relative velocity between meshing gearsat contact points.
It therefore can be concluded that
C(q,q)q = [ Chq
C] q
= โ[[
(Chqq)q q + 2Ch
q๐กq + C๐ก๐ก(Cq)q q + (๏ฟฝcq + C๐ก) q + ๏ฟฝc๐ก
]].
(7)
Thus, (2) can be rewritten as
Q๐ = [Qh๐
Qnh๐
] = โ[[
(Chqq)q q + 2Ch
q๐กq + C๐ก๐ก(Cq)q q + (๏ฟฝcq + C๐ก) q + ๏ฟฝc๐ก
]]. (8)
It should be pointed out that the nonholonomic constraintequations given in (4) must be nonintegrable and cannotbe reduced to an integrable form like the ones of geometricconstraints of other joints.
In this case, the case of multibody system involvingholonomic and nonholonomic constraints, the equations ofmotion can be expressed as
Mq = Q +Q๐ +Q๐ก, (9)
where the Q vector includes all external and elastic forces,Q๐ are the generalized reaction forces due to holonomicconstraints, and Q๐ก are the generalized forces due to thenonholonomic constraints. These forces can be written asfollows:
Q๐ = โCh๐q ๐
h
Q๐ก = โCnh๐q ๐
nh. (10)
4 Journal of Energy
2.1. Constraints Forces. In this section, a procedure for deter-mining the reaction forces associated with the generalizedcoordinates of constrained bodies is presented. These reac-tions are mapped into the Cartesian coordinates in the localand global domain as well. In spatial analysis, joints thatconnect two adjacent bodies have forces and moments at thejoint definition points defined in the Cartesian space. Theseconstraint forces significantly affect the dynamic stressesbecause these forces are heavily influenced by the impul-sive contact forces. In the next subsections, the vector ofLagrange multipliers is evaluated corresponding to specifiedholonomic and nonholonomic constraints, and, accordingly,the reaction forces equations are derived in the form suitablefor design process.
2.1.1. Holonomic Constraints Forces. The generalized con-straint forces acting on rigid body ๐, as a result of certain jointwith body ๐, can be written in terms of the Lagrangemultipliers associated with the joint as
Q๐๐ = C๐q๐๐h = [C๐R๐
C๐๐๐
] ๐h. (11)
The virtual work of these generalized constraint forces canthen be expressed as
๐ฟ๐๐๐ = Q๐๐๐ ๐ฟq๐ = (C๐R๐๐h)๐ ๐ฟR๐ + (C๐๐๐๐h)๐ ๐ฟ๐๐. (12)
In order to calculate the joint reaction forces associated withthe Cartesian coordinates, let us assume that F๐๐ and M๐๐ be,respectively, the vectors of actual joint reaction forces andmoments acting on body ๐ as a result of the certain (revolute,prismatic, etc.) joint with body ๐. The virtual work of thesereaction forces and moments can be written as
๐ฟ๐๐๐ = F๐๐๐ ๐ฟr๐๐ +M๐๐๐ ๐ฟฮจ๐๐, (13)
where r๐๐ (see Figure 2) is the position vector of the point ofapplication (say point ๐) of the force F๐๐. ฮจ
๐๐ is the absolute
rotation vector of the body local coordinate system. Thevector r๐๐ can be expressed as r๐๐ = R๐ + A๐u๐ and thereforethe vectors ๐ฟr๐๐ and ๐ฟฮจ๐๐ can be expressed in terms of thegeneralized coordinates of body ๐ as
๐ฟr๐๐ = ๐ฟR๐ โ A๐u๐G๐๐ฟ๐๐
๐ฟฮจ๐๐ = A๐G๐๐ฟ๐๐, (14)
whereu๐ is the local position vector of the point of applicationof the force F๐๐, defined in the body coordinate system, andvirtual rotation is defined as ๐ฟฮจ = [๐ฟ๐๐ฅ ๐ฟ๐๐ฆ ๐ฟ๐๐ง]๐ of thebody local coordinate system. Substituting (14) into (13), oneobtains
๐ฟ๐๐๐ = F๐๐๐ ๐ฟR๐ + (M๐๐๐ A๐G๐ โ F๐๐๐ A๐u๐G๐) ๐ฟ๐๐. (15)
Comparing (12) with (15), one obtains
F๐๐ = C๐R๐๐h, (16)
C๐๐๐๐
h = (M๐๐๐ A๐G๐ โ F๐๐๐ A๐u๐G๐)๐ . (17)
Substituting the value of F๐๐ of (16) into (17), one can concludethat
M๐๐ = A๐ (14G๐C๐๐๐ + A๐u๐
๐ A๐๐C๐R๐)๐h. (18)
Equations (16) and (18) represent the necessary forms toevaluate the reaction forces and reactionmoments for certainkinematic constraints.
2.1.2. Nonholonomic Constraints Forces. The virtual work ofthese generalized (nonholonomic) constraint forces, Q๐ก, canthen be written as
๐ฟ๐๐๐ก = Q๐๐๐ก ๐ฟq๐ = (Cnh๐R๐ ๐
nh)๐ ๐ฟR๐ + (Cnh๐๏ฟฝ๏ฟฝ๐ ๐
nh)๐ ๐ฟ๐๐. (19)
In order to map Q๐ก, into the Cartesian coordinates, let usassume that F๐๐ก is the vector of forces acting on body ๐; thevirtual work of these forces can be written as
๐ฟ๐๐๐ก = F๐๐๐ก ๐ฟr๐๐ก, (20)
where r๐๐ก is the position vector of the tangential point, at whichthe nonholonomic constraints are formulated. Recall that thevirtual displacement ๐ฟr๐๐ก can be presented as
๐ฟr๐๐ก = ๐ฟR๐ โ A๐u๐กG๐๐ฟ๐๐. (21)
One obtains
๐ฟ๐๐๐ก = F๐๐๐ก ๐ฟR๐ โ F๐๐๐ก A๐u๐กG๐๐ฟ๐๐. (22)
Comparing (19) with (22) yields
F๐๐ก = Cnh๐R๐ ๐
nh, (23)
Cnh๐๏ฟฝ๏ฟฝ๐ ๐
nh = โ (F๐๐๐ก A๐u๐กG๐)๐ = โG๐๐u๐๐ก A๐๐F๐๐ก. (24)
The following equation can be obtained:
14A๐G๐Cnh๐๏ฟฝ๏ฟฝ๐ ๐
nh = โu๐๐ก F๐๐ก. (25)
Thus, the resulting moment vector in the global frame can beconcluded as
M๐๐ก = โ14A๐G๐Cnh๐๏ฟฝ๏ฟฝ๐ ๐
nh. (26)
In the local coordinate system, one can obtain the followingequations:
F๐๐ก = A๐๐Cnh๐R๐ ๐
nh (27)
M๐๐ก = โ14G๐Cnh๐๏ฟฝ๏ฟฝ๐ ๐
nh. (28)
Equations (23) and (27) represent the resulting forces, while(26) and (27) represent the forms of the resulting momentsdue to nonholonomic constraints in global and local frames,respectively.
Journal of Energy 5
2.1.3. Position and Velocity Stabilization. After each integra-tion step, some constraint drifting at the position and velocitylevels may take place. Therefore after each integration steprelying on acceleration, it is necessary to move positions andvelocities back to their manifolds. This corrections process iscalled poststabilization procedure and can be implementedusing Newton-Raphson method [17]. The poststabilizationof rotational multibody system subjected to nonholonomicconstraints has been presented recently in [18].
3. DFD Procedure
Dynamics for Design (DFD) is the integration of recentadvances in systemdynamics, including nonlinearities, vibra-tion analysis, and multibody systems with current designmethodologies. The goal of integrating these subjects in oneprocedure is to obtain an efficient design cycle in terms ofimproved system reliability and consistent behavior acrossoperating environments.
WTDT components include the rotor blades, gearbox(one, two, or three stages), and electrical generator. The pur-pose of the gearbox is to transfer the low rotational speed ofthe rotor to relatively high speed required for the generator.Consequently, the design process starts by assigning the pre-liminary size of the blade structure and from the other side bycalculating the pressure distribution along the blade [7] andthus obtains the induced lift and drag forces. Mapping theseforces from the blade coordinate system (frame) into the rotorframe enables the designer to calculate the generalized forcesapplied along the blade structure. Introducing the size ofthe blade as well as the generalized forces into the flexiblemultibody dynamics module based on continuum modelof blade enables calculation of the induced torque and theassociated rotational speed of the rotor blades. As shown inFigure 3, based on the outputs of blade dynamics module, thedesigner can assign the preliminary configuration and size ofthe required gearbox. It should bementioned that small windturbines are equipped with parallel-shaft gear trains, whichare commercially available from numerous manufacturers.On the other hand, in larger wind turbines, the planetarydesign (epicyclic configuration) is definitely dominating. Foroutputs of several megawatts, two or three planetary stagesare used within one or two parallel stages [19]. This stephelps in calculating the geometrical dimensions and inertiaparameters of the gearbox as multibody system.
Based on the multibody system approach, described inSection 2, and according to the preliminary configurationselected for the gearbox, the designer can calculate the reac-tion forces (between gears and ground) and contact forcesbetween the gears. This step will be presented clearly duringpractical case study, presented in Section 4.
Once the reactions and contact forces are calculated,these forces can be mapped into components in the normal,tangential, and axial directions. Accordingly, the normal aswell as shear stresses can be evaluated. Checking the stressesof the gears is carried out according to the AmericanGear Manufacturers Association (AGMA) [20], which hasthe responsible authority for the dissemination of stan-dards pertaining to the design and analysis of gearing [20].
Resistance to tooth breakage is normally dependent uponthe bending stress occurring at the contact area of matingteeth. These stresses are the key start of the design processby which trade-offs optimization can be carried out betweenmaterial selection and the related dimensions of designedcomponents.
In this section, two subsections are introduced in order tocomplete the picture in its general form. The first subsectionpresents the necessary equations of estimating the general-ized forces induced due to the aerodynamic loads. The othersubsection presents the necessary equations to calculate andexamine the stresses according to AGMA code.
3.1. Aerodynamic Loads. The aerodynamic loads resultingfrom the attacking wind stream consist of two main compo-nents. The first component is called drag force, which comesparallel to the relative wind velocity. The second componentis called lift force, and its direction is perpendicular torelative wind velocity. Based on the numeric values of theangle of attack ๐ผ, as illustrated in Figure 4, Reynolds andMach number, the blade coefficients can be identified. Thesecoefficients are the pressure distribution across the blade,๐ถP(๐ฅ), lift coefficient, ๐ถL, and the drag coefficient, ๐ถD, andcan be estimated for specific geometric shape of turbine blade[7, 21].
As shown in Figure 4, the relative wind velocity ๐ isalways greater than the upstream velocity k2 at the rotor planeby the vectorial sumof tangential velocityu; that is,๐ = k2+u.This tangential velocity is induced from the rotorโs angularspeed and thus depends on the local distance along the ๐ฆ-axis on the blade frame; that is, u = u(๐ฆ), and, consequently,there is a relative velocity for each cross section of the blade;that is, ๐ = ๐(๐ฆ). The equations of lift and drag forces can bewritten in following forms [22]:
๐นL (๐ฆ) = 12๐ [๐ (๐ฆ)]2 ๐ด [๐ถL (๐ผ)]๐นD (๐ฆ) = 12๐ [๐ (๐ฆ)]2 ๐ด [๐ถD (๐ผ)] ,
(29)
where ๐ is the air density,๐ด is the projected area, and (๐๐2/2)is the dynamic pressure.These forces are defined with respectto the blade frame; accordingly, they should be mappedinto rotor frame, which can be carried out by the followingtransformation:
F๐P๐ = [[[[
๐น๐ฅ๐น๐ฆ๐น๐ง]]]]= [[[
0 0โ sin๐ผ cos๐ผcos๐ผ sin๐ผ
]]][[[[
๐น๐๐L (๐ฅ)๐๐๐๐นD (๐ฅ)๐๐๐]]]]. (30)
Thus, the generalized forces can be estimated by accumulat-ing these terms along the blade span as follows [23]:
(Q๐๐)๐ = A๐F๐P๐ = A๐ โซ๐0F๐(๐ฆ=๐/4)๐ ๐๐ฅ
(Q๐๐)๐ = โG๐๐โซ๐0๐ขP๐๐F๐(๐ฆ=๐/4)๐ ๐๐ฅ
(Q๐๐)๐๐ = โซ๐0S๐๐P F๐(๐ฆ=๐/4)
๐ ๐๐ฅ.(31)
6 Journal of Energy
Demanded power
Assign โpreliminaryโsize of wind turbine
(i) Number of blades
(ii) Blade structure
(iii) Rotor diameter
(iv) Height
Flexible multibodydynamics
Continuum blademodel
Wind speed
(i) Pressure distribution (ii) Li๏ฟฝ & drag forces (iii) Generalized forces on
blade structure
Aerodynamicforce model
(i) Masses
(ii) Moment of inertia(i) Geometrical dimensions
Rigid multibody dynamics
Holo., nonholonomic system constraints
(i) Reaction forces(ii) Contact forces
Design gear sizing(i) Number of teeth, face width
(ii) Module, outer and inner diameters
(i) Generalized forces on rotor body (i) Cut-o๏ฟฝ speed
(ii) Maximum speed
Assign โpreliminaryโ size of gearbox
(i) Number of stages
(ii) Type of each stage (parallel & planetary)
(iii) Gear ratio (for each stage)
(iv) Pressure angel (for each gear)
Materiallibrary
Material
properties
Change materialif possible?
Applying AGMA
code to check bending
and contact stresses
No
Yes
Check
Stop
Mod
ify si
ze o
f the
gea
rbox
safety factorโฅ 1
Figure 3: DFD procedure for wind turbine gearbox; the dashed shaded part denotes the blade dynamics module, which is not the focus ofthis paper, and the solid shaded part denotes the gearbox dynamics module, which is the main issue concerned in this investigation.
Journal of Energy 7
Chord
line
FL
FD
x z
F๏ผ๏ผข๏ผฏ๏ผฌ๏ผญ๏ผฎRotor
Bladeframe
frame
ฮฉZ
u
2
2
y
//
// 2 // z
F๏ผ๏ผฉ๏ผฌ๏ผซ๏ผฏ๏ผ
Figure 4: Aerodynamic forces on blade section.
These forces are introduced to the blade structuremodulein order to estimate the induced torque and the associatedrotational speed of the rotor blades.
3.2. Stress Analysis. The driver gear transmits motion todriven gear at contact point at pressure angel with respectto the contact line. This force can be calculated by obtainingthe nonholonomic Lagrangemultipliers, which is determinedusing the rigid body simulation of the model. It can be ana-lyzed into two components, tangential and normal (radial)components. According to AGMA standard, the tangentialcomponent is the source of bending and contact stressesacting on gear tooth and these stresses are used to establishdesign or check safety for realistic design.
Under the assumptions stated in [24] and according to theflowchart shown in Figure 3, the stress analysis can be carriedout in the postprocessing for design stage. The bending andcontact stresses are estimated according to AGMA equations[20, 25]. The estimated stresses are compared with allowablestresses that gears materials can withstand; this ratio is calledthe factor of safety, SF, and it must be greater than one;otherwise the dimensions of gears should bemodified and/orother material can be selected. The basic equations andterminology of bending and contact stresses according toAGMA are stated in the following paragraph.
The bending stress can be estimated as
๐Bending = ๐๐ก๐พ0๐พV๐พ๐ ๐พ๐๐พ๐ต๐น๐๐ฝ , (32)
where ๐๐ก is tangential force component, ๐พ0 is the overloadfactor, ๐พV is the dynamic factor, ๐พ๐ is the size factor, ๐พ๐is the load distribution factor, and ๐พ๐ต is the rim thicknessfactor. Also, ๐น is facewidth,๐ is module of the gear, and ๐ฝ isgeometry factor on bending. Equation (32) is used to calculateactual bending stress acting on gear teeth.This value of stressis compared with allowable bending stress. The allowablebending stress can be estimated by
๐๐ต-allowable = ๐๐ก๐๐๐พ๐๐พ๐ , (33)
where ๐๐ก is the allowable bending stress (from materialstables), ๐๐ is the bending stress cycle factor, ๐พ๐ is the relia-bility factor, and ๐พ๐ is the temperature factor.
Furthermore, the actual contact stress can be estimated asfollows:
๐contact = ๐ถ๐โ๐๐ก๐พ0๐พV๐พ๐ ๐พ๐๐น๐ท๐ผ , (34)
where ๐ถ๐ is the elastic coefficient factor, ๐ท is the pitchdiameter of gear, and ๐ผ is the geometry factor on contact.The other values are defined previously in the bending stressequation. The allowable contact stress can be estimated as
๐๐ถ-allowable = ๐๐๐๐๐ถ๐ป๐พ๐๐พ๐ , (35)
where ๐๐ is the allowable contact stress (from gears materialstables), ๐๐ is the contact stress cycle factor, and ๐ถ๐ป is the
8 Journal of Energy
0 10 20 30 40 50 60
Time (s)
โ5
โ4
โ3
โ2
โ1
0
1
2
3
4
5
6
7
โ5โ4โ3โ2โ101234567
Torq
ue (M
Nยทm
)
0 1 2 3 4 5 6 7 8 9 10
Figure 5: Rotor-blade exerted torque.
Table 1: Size of wind turbine.
Item ValueNumber of blades 3Blade length [m] 41Hub diameter [m] 3.7Blade Youngโs modulus [MPa] 100Blade density [Kg/m3] 2000
hardness ratio factor. By using (32)โ(35), the safety factors canbe estimated for bending and contact stresses as follows:
SFB = ๐Bending๐๐ต-Allowable (36)
SFC = ๐contact๐๐ถ-Allowable . (37)
4. Case Study of Practical Wind Turbine
A case study of 1.4MW nominal power is introduced inthis section in order to evaluate the proposed procedure forWTDT design.The turbine is selected with the specificationslisted in Table 1.
The nominal power is attained at wind speed of 10 [m/s],by which cut-in/cut-out speeds can be proposed as 4 [m/s]and 20 [m/s], respectively. By considering a constant windspeed of 10 [m/s], using the blade dynamics module (seeFigure 3), the steady state value of the exerted torque on therotor-blade is estimated as 1 [MNโ m] (see Figure 5), and theassociated rotational speed is found to be 14.2 [rpm] (seeFigure 6).The high oscillatory nature shown in Figures 5 and6 results from the large flexibility of blades structure. It ispresumable that this oscillation will be damped out when therotor hub is attached to the gearbox; this is because of the
damping effect between rotating components and the wallstructure [26].
4.1. Preliminary Gearbox Design. Planetary gearing systemsintroduce higher ratios of power transmission than parallelaxis gears and are able to provide high-speed ratios within acompact volume. However, the inaccessibility of their com-ponents and high loads on the shaft bearings are consideredone of the most important difficulties of maintenance andoperations. The standard gearbox configurations include oneor two planetary stages followed by one or two parallel stages.As is common in the horizontal wind turbines, the ring gearwould be connected to the bedplate, and the planet carrierwould be connected to the rotor hub, while the sun gearwould be connected to the output shaft, which is connectedto the next stage.
In the case study under consideration, standard gearboxconfiguration is selected such that one planetary stage andtwo parallel stages are connected in series. The overall speedratio should be in range of 85 to 100 in order to attain thenominal speed of the electric generator. The preliminary sizeand specifications can be selected as listed in Table 2. In theplanetary stage, the center distance incremental factor is heldat zero, and thus the geometric relation of ๐1 = (๐2 + 2๐5) isvalid, where ๐1, ๐2, and ๐5 are the radii of ring, sun, and planetgears, respectively.
4.2. Multibody System and Constraints. As shown in Table 2,the three stages can be constructed using thirteen bodies;the ring gear is held to be fixed and considered as thewall structure (ground). The planetary stage consists of ninebodies (see Figure 7); three pins are considered to connectthe carrier body with planet gears. The multibody system ofthe planetary stage can be constructed with the arrangementlisted in Table 3.The total number of generalized coordinatesis 63; the dependency between them can be determined by
Journal of Energy 9
0 10 20 30 40 50 60
Time (s)
โ4
โ2
0
2
4
6
8
10
12
14
16
18
20
Ang
ular
velo
city
(RPM
)
0 1 2 3 4 5โ202468
101214
Figure 6: Rotor-blade rotational speed.
Table 2: Preliminary size of the planetary stage.
Item ValueStage (1): planetary stage
Gear ratio 6Number of planets 3Number of teeth on the sun 24Number of teeth on the planet 48Number of teeth on the ring 120
Stage (2): parallel stageGear ratio 3.6Number of teeth on the input gear 83Number of teeth on the output gear 23
Stage (3): parallel stageGear ratio 3.96Number of teeth on the input gear 103Number of teeth on the output gear 26
defining the constraints function according to the type ofjoints.
The holonomic constraints can be considered as listedin Table 4; the total number of holonomic constraints is 58;therefore, the system has 5 degrees of freedom. It shouldbe noted here that every revolute joint is accompanied withsignificant damping proportion, between the carrier, sun, andthe ground and between the pins and planet gears.
The nonholonomic constraints result from the equaltangential velocity of themeshing gears at their contact pointsand therefore zero relative velocity between them. Thus, thefollowing equations can be applied:
๐3๐3๐ง = ๐2๐2๐ง + ๐5๐5๐ง๐1๐1๐ง = ๐3๐3๐ง + ๐5๐5๐ง, (38)
where ๐3 = (๐2 + ๐5) is the center distance and ๐1 = (๐2 +2๐5) is the radius of the ring gear. By considering no profileshifting, that is, the center distance incremental factor is zero,and using the definition of gear module, that is, ๐ = ๐ท/๐,where ๐ท is the pitch diameter and ๐ is the number of teeth,thus (38) can be rewritten as follows:
๐2 (๐2๐ง โ ๐3๐ง) + ๐5 (๐5๐ง โ ๐3๐ง) = 0(๐2 + 2๐5) ๐1๐ง โ (๐2 + ๐5) ๐3๐ง โ ๐5๐5๐ง = 0. (39)
Two more constraints equations can be added for the othertwo planet gears, to relate the rotational speed of the carrier,planet, and sun gears, such that
๐2 (๐2๐ง โ ๐3๐ง) + ๐7 (๐7๐ง โ ๐3๐ง) = 0๐2 (๐2๐ง โ ๐3๐ง) + ๐9 (๐9๐ง โ ๐3๐ง) = 0. (40)
The total number of constraints, the holonomic and non-holonomic constraints, became 62; thus the system has onlyone degree of freedom. By applying the exerted torque (seeFigure 5) on the carrier body along with the rotational axis(local ๐ง3-axis as shown in Figure 7), the dynamic simulationof the planetary stage system shows that the system angularvelocities obey the required ratios. As shown in Figure 8, theangular velocity of the carrier body is 14.2 [rpm] and theangular velocity of the sun gear is 85.2 [rpm] with speed ratioof 6.
The simulation results of the multibody system repre-sented by (1) include the states vector (generalized acceler-ation and velocities) plus a vector of 62 Lagrange multipliers.Some of these multipliers are plotted in Figures 9 and 10. Thenumber of Lagrange multipliers associated with holonomicconstraints is 58 multipliers; among them, there are 9 trivialmultipliers. These trivial multipliers represent the Eulerparameters constraints ๐๐๐ = 1, for each body. The other 49
10 Journal of Energy
Table 3: Body index of planetary stage.
Bodynumber
Componentname
Index of gen.coordinates
Inertia properties๐[Kg] ๐ผ๐ฅ๐ฅ [Kgโ m2] ๐ผ๐ฆ๐ฆ [Kgโ m2] ๐ผ๐ง๐ง [Kgโ m2]1 Ring 1โ7 1238 556.639 556.639 1083.485
2 Sun 8โ14 263.5 5.5885 5.5885 4.83703 Carrier 15โ21 405.091 44.2734 44.2734 88.2644 Pin (1) 22โ28 254.3 46.14 46.14 1.875 Planet (1) 29โ35 1113.1 52.284 52.284 77.786 Pin (2) 36โ42 254.3 46.14 46.14 1.877 Planet (2) 43โ49 1113.1 52.284 52.284 77.788 Pin (3) 50โ56 254.3 46.14 46.14 1.879 Planet (3) 57โ63 1113.1 52.284 52.284 77.78
y1
z1 x1
y5
z5x5y4
z4x4
y6
z6 x6y3
z3 x3y2
z2 x2
y8
z8 x8
y9
z9 x9
y7
z7x7
Z
Y
X
Figure 7: Multibody system model of planetary stage, stage (1).
multipliers can be used to calculate the reaction forces andmoments by using (16) and (18).
Figure 9 shows the nonzero Lagrange multipliers associ-ated with the first joint, that is, the rigid joint between thering gear and the ground. The first and second Lagrangemultipliers are zero, because they are related to the forcesalong ๐ฅ- and ๐ฆ-axes; however, the third multiplier is relatedto the force along the ๐ง-axis, which is nonzero value at allcases because of the gear weight. The other multipliers arerelated to the reaction moments and associated with twoEuler parameters ๐15 and ๐16 .The reaction forces andmomentsof the ring gear are plotted in Figures 11 and 12; note that thevalues are considered with respect to the global coordinatesystem (see Figure 7).
Figure 10 shows the Lagrange multipliers associated withthe nonholonomic constraints, that is, (39)-(40). Since thesemultipliers are related to the angular velocities of the rotatinggears and the ring, the resulting forces estimated by (23) arezeros; however, (26) and (28) give the driving torques of therotating gears. Thus, the driving forces between gears canbe obtained for the planet, sun, and also the ring gears. Thetangential force between the sun and planet gear is plottedin Figure 13, with a steady state value of 412 [KN]. It shouldbe noted that arbitrary values are selected for the pressureangle, module, center distance, and the facewidth of thematching gears. These values are taken as follows: ๐ = 24.5โ,๐ = 15 [mm], ๐3 = 540 [mm], and ๐น = 380 [mm]. Thesenumerical values are used to estimate the inertia properties
Journal of Energy 11
Table 4: Holonomic constraints of stage (1).
Joint Type Body (๐) Body (๐) Number of constraintsRigid Ground Ring 7Revolute Ground Sun 6Revolute Ground Carrier 6Rigid Carrier Pin (1) 7Revolute Pin (1) Planet (1) 6Rigid Carrier Pin (2) 7Revolute Pin (2) Planet (2) 6Rigid Carrier Pin (3) 7Revolute Pin (3) Planet (3) 6
0 10 20 30 40 50 60
Time (s)
โ40โ30โ20โ10
0102030405060708090
Ang
ular
velo
city
(RPM
)
๏ผ๏ผฃ๏ผจ๏ผกz
๏ผ๏ผ๏ผฌ๏ผฌ๏ผฃ๏ผ๏ผฌz
๏ผ๏ผฆ๏ผ๏ผจ๏ผ๏ผฎz
๏ผ๏ผฏ๏ผจz
Figure 8: Angular speed of gears in planetary stage.
0 10 20 30 40 50 60
Time (s)
โ600โ500โ400โ300โ200โ100
0100200300400500600
Lagr
ange
mul
tiplie
rs G
(1)
3
5
6
Figure 9: Nonzero Lagrange multipliers due to fixed ring gear.
of Table 3 by aid of CAD software. For the preliminary designand according to the tangential force estimated, it is foundthat the safety factor on contact between the sun and planetgears is SFC2,5 = 1.5 and between the planet and ring gear isSFC5,1 = 1.65. Also, the minimum safety factor on bending at
0 10 20 30 40 50 60
Time (s)
โ3
โ2
โ1
0
1
2
3
4
5
Lagr
ange
mul
tiplie
rs (K
)
๏ผจ๏ผข1
๏ผจ๏ผข2
๏ผจ๏ผข3
๏ผจ๏ผข4
Figure 10: Lagrange multipliers of nonholonomic constraints.
0 10 20 30 40 50 60
Time (s)
0123456789
10111213
Reac
tion
forc
es (K
N),
glob
al, r
ing
gear
F๏ผ๏ผฆx
F๏ผ๏ผฆy
F๏ผ๏ผฆz
Figure 11: Reaction forces in global coordinates of ring gear.
0 10 20 30 40 50 60
Time (s)
โ500
โ400
โ300
โ200
โ100
0
100
200
300
400
500
M๏ผ๏ผฆx
M๏ผ๏ผฆy
M๏ผ๏ผฆz
Reac
tion
mom
ents
(KN
m),
glob
al, r
ing
gear
Figure 12: Reaction moments in global coordinates of ring gear.
12 Journal of Energy
Table 5: Working designs of planetary stage.
# ๐3 ๐ ๐2 ๐5,7,9 ๐1 ๐ SF๐ถ2,5 SF๐ถ5,1 min (SF๐ต)๐ min (SF๐ต)๐1 420 5 53 115 โ283 6.34 1.58 1.66 0.54 1.282 420 5 54 114 โ282 6.22 1.58 1.66 0.55 1.303 420 5 55 113 โ281 6.11 1.58 1.66 0.56 1.324 480 5 61 131 โ323 6.30 1.59 1.66 0.62 1.475 480 5 62 130 โ322 6.19 1.59 1.66 0.63 1.496 480 5 63 129 โ321 6.10 1.59 1.66 0.64 1.517 540 5 68 148 โ364 6.35 1.60 1.66 0.69 1.658 540 5 69 147 โ363 6.26 1.60 1.66 0.70 1.669 540 5 70 146 โ362 6.17 1.60 1.66 0.71 1.6810 540 5 71 145 โ361 6.09 1.60 1.66 0.72 1.7011 600 5 76 164 โ404 6.32 1.61 1.66 0.77 1.8412 600 5 77 163 โ403 6.23 1.61 1.66 0.78 1.8513 600 5 78 162 โ402 6.15 1.61 1.66 0.79 1.8714 600 5 79 161 โ401 6.08 1.61 1.66 0.80 1.8915 540 6 57 123 โ303 6.32 1.59 1.66 0.83 1.6516 540 6 58 122 โ302 6.21 1.59 1.66 0.84 1.6717 540 6 59 121 โ301 6.10 1.59 1.66 0.86 1.6918 420 7 38 82 โ202 6.32 1.55 1.66 0.75 1.2719 420 7 39 81 โ201 6.15 1.56 1.66 0.77 1.3020 420 8 33 72 โ177 6.36 1.54 1.66 0.85 1.2621 420 8 34 71 โ176 6.18 1.54 1.66 0.87 1.2922 480 8 38 82 โ202 6.32 1.55 1.66 0.98 1.4523 480 8 39 81 โ201 6.15 1.56 1.66 1 1.4824 540 8 43 92 โ227 6.28 1.57 1.66 1.10 1.6425 540 8 44 91 โ226 6.14 1.57 1.66 1.13 1.6726 600 8 47 103 โ253 6.38 1.58 1.66 1.21 1.8027 600 8 48 102 โ252 6.25 1.58 1.66 1.23 1.8428 600 8 49 101 โ251 6.12 1.58 1.66 1.25 1.8729 540 9 38 82 โ202 6.32 1.56 1.66 1.23 1.6330 540 9 39 81 โ201 6.15 1.56 1.66 1.26 1.6731โ 420 10 27 57 โ141 6.22 1.52 1.65 1.07 1.2732 480 10 31 65 โ161 6.19 1.53 1.65 1.23 1.4633 540 10 34 74 โ182 6.35 1.54 1.66 1.36 1.6234 540 10 35 73 โ181 6.17 1.55 1.66 1.39 1.6535 600 10 38 82 โ202 6.32 1.56 1.66 1.51 1.8136 600 10 39 81 โ201 6.15 1.56 1.66 1.55 1.8537 540 12 29 61 โ151 6.21 1.53 1.65 1.65 1.6338โโ 540 15 24 48 โ120 6 1.50 1.65 2.10 1.6639 540 15 23 49 โ121 6.26 1.50 1.65 2.02 1.60๐3: center dist.;๐: module; ๐: # of teeth;๐: speed ratio; SF: safety fac. โOptimal design. โโPreliminary design.
the gear root is 2.1; that is, min(SFB,1, SFB,2, SFB,5)๐ = 2.1, andat the gear flank it is 1.66; that is, min(SFB,1, SFB,2, SFB,5)๐ =1.66. These factors can be accepted as working design for thecurrent gearing configuration.
5. Design Procedure
The design procedure assumes zero profile shifting coef-ficients for the planetary stage. These coefficients can be
selected in order to obey the geometrical boundaries as wellas to achieve some optimum values for the sliding velocitiesand safety factors. Asmentioned earlier, themultibodymodelconstructed in this paper concerns only the case of perfectcenter distances and, therefore, these coefficients are excludedfrom the optimization process. Based on the acceptableranges of gearing ratios, center distances, and the gearingnormal modules, 39 workable designs are found and listed inTable 5.
Journal of Energy 13
0 10 20 30 40 50 60
Time (s)
โ100โ50
050
100150200250300350400450500
Tage
ntia
l for
ce (K
N)
Figure 13: Tangential force due to nonholonomic constraints.
4 5 6 7 8 93 11 12 13 14 15 16 17 18 1910
Module (mm)
390
410
430
450
470
490
510
530
550
570
590
610
Cen
ter d
istan
ce (m
m)
#11
#7 #15 #24 #29 #33
#4 #22 #32
#1 #18 #20 #31
#37 #38
#26 #35
Figure 14: Center distance of workable designs of planetary stage.
Figure 15 shows the workable designs which satisfy acenter distance between 420 [mm] and 600 [mm]. Figure 14shows the workable designs that satisfy the required rangeof speed ratios. As stated in Section 4.1, the speed ratioof the gearbox is accepted for values between 85 and 100;therefore, the speed ratio of the planetary stage should beaccepted in the range of 6 to 6.7. By the use of the multibodymodel (see Figure 3), the tangential force is calculated forall workable designs listed in Table 5. The facewidth isreestimated according to AGMA; that is, ๐น = (3โผ5)๐๐ [20].The inertia properties are updated and then the contactand bending stresses and the corresponding safety factorsare calculated. The accepted workable designs based on thesafety factor on contact between the sun and planet gear andbetween the planet and ring gears are plotted in Figures 16and 17. Similarly, the accepted workable designs based onthe minimum safety factor on bending at the gears rootsand flanks are illustrated in Figures 18 and 19. The centerdistances, speed ratios, and safety factors are plotted againstthe normal module, which comes in values between 5 and15 [mm].
The trade-offs optimization is a logical operation (AND/OR/NOT) among all workable designs. This operation isbased on the required center distance ๐3, speed ratio ๐, andacceptable safety factors SF. Practically, it can be carriedout using the results shown in Figures 14โ19. The optimal
Spee
d ra
tio #7#1 #15#18#22 #29
#2 #16#19#3 #17 #25
#38
#21 #30 #31 #37
#35 #39#20 #33
4 5 6 7 8 93 11 12 13 14 15 16 17 18 1910
Module (mm)
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Figure 15: Speed ratio of workable designs of planetary stage.
#1 #15 #18#22 #29 #35#20 #31 #37 #38
4 5 6 7 8 93 11 12 13 14 15 16 17 18 1910
Module (mm)
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Con
tact
safe
ty fa
ctor
(sun
/pla
net)
Figure 16: Safety factor on contact between sun and planet gears.
design, #๐ท, is computed according to the following formu-la:
(๐3 โช ๐ โช๐ โช SF) = #๐ท. (41)
For instance, if the required center distance is 600 [mm],speed ratio is 6.3, and normal module is 10 [mm], (41) gives(600 โช 6.3 โช 10 โช (SF > 1)) = #35. The design number 35 isthe only design that satisfies all criteria, and the safety factorsare greater than 1.5 (see Table 5).
If minimum center distance is required, with acceptablesafety factors (SF > 1), whatever the values of the moduleand speed ratio, the solution is (min(๐3) โช (SF > 1)) = #31.This solution gives a center distance of 420 [mm]with a speedratio of 6.22, which lies in the required range, and module of10 [mm]. In Table 5, design number 31 refers to an optimaldesign for compact size, precise speed ratio, and acceptablefactors of safety.
The previous systematic procedure can be applied fordesigning the other parallel axis stages with less computa-tional effort than the planetary stage.
6. Summary and Conclusions
Dynamics for Design (DFD) is the integration of recentadvances in system dynamics including nonlinearities, vibra-tion analysis, and multibody systems with current designmethodologies. This paper proposed a design procedure for
14 Journal of Energy
#11#7#4#1
#15
#19
#26#24#23#22#20
#29
#31#32#33#34#35
#37
#38
#18
4 53 7 8 96 11 12 13 14 15 16 17 18 1910
Module (mm)
1.4
1.45
1.5
1.55
1.6
1.65
1.7
Con
tact
safe
ty fa
ctor
(pla
net/r
ing)
Figure 17: Safety factor on contact between ring and planet gears.
#26
#23
#24#25
#27#28 #30
#29
#31
#32
#33#34
#35#36
#37
4 5 6 7 8 93 11 12 13 14 15 16 17 18 1910Module (mm)
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Bend
ing
safe
ty fa
ctor
Figure 18: Minimum safety factor on bending at root.
the wind turbine drive-train (WTDT). The design procedureis based on the multibody system (MBS) approach, whichis used to estimate static as well as dynamic loads based onthe Lagrange multipliers. The paper presented the necessaryderivation of Lagrangemultipliers associatedwith holonomicand nonholonomic constraints assigned for the drive-trainmodel. Based on the dynamic model and the obtained forces,the design process of a planetary gear stage is implemented
with trade-offs optimization in terms of gearing parameters.Awind turbine of 1.4Megawatts nominal power is introducedas an evaluation study of the proposed procedure, in whichthe main advantage is the systematic nature of designingsuch complex system in motion. Finally, the nonholonomicconstraints of the engaged gears presented in this paperassume zero profile shifting coefficients between them. Thisissue will be considered in future work to obey the allowable
Journal of Energy 15
#14#13#12 #11
#10
#15
#18 #20#21
#22#23
#24#25
#19
#26
#29
#31
#32
#33#34
#35
#36
#37#38
#39
#30
#27#28
#16#17#9
#8
#6#5
#4
#3#2#1
#7
4 5 6 7 8 93 11 12 13 14 15 16 17 18 1910Module (mm)
1.3
1.4
1.5
1.6
1.7
1.8
1.9Be
ndin
g sa
fety
fact
or
Figure 19: Minimum safety factor on bending at flank.
stresses and the geometrical boundaries of the drive-train aswell.
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
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